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Theorem resspsradd 18575
Description: A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
resspsr.s  |-  S  =  ( I mPwSer  R )
resspsr.h  |-  H  =  ( Rs  T )
resspsr.u  |-  U  =  ( I mPwSer  H )
resspsr.b  |-  B  =  ( Base `  U
)
resspsr.p  |-  P  =  ( Ss  B )
resspsr.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
Assertion
Ref Expression
resspsradd  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )

Proof of Theorem resspsradd
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 resspsr.u . . 3  |-  U  =  ( I mPwSer  H )
2 resspsr.b . . 3  |-  B  =  ( Base `  U
)
3 eqid 2429 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
4 eqid 2429 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
5 simprl 762 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
6 simprr 764 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
71, 2, 3, 4, 5, 6psradd 18541 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X  oF ( +g  `  H
) Y ) )
8 resspsr.s . . . 4  |-  S  =  ( I mPwSer  R )
9 eqid 2429 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2429 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
11 eqid 2429 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
12 fvex 5891 . . . . . . . 8  |-  ( Base `  R )  e.  _V
13 resspsr.2 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubRing `  R
) )
14 resspsr.h . . . . . . . . . . 11  |-  H  =  ( Rs  T )
1514subrgbas 17952 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
1613, 15syl 17 . . . . . . . . 9  |-  ( ph  ->  T  =  ( Base `  H ) )
17 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1817subrgss 17944 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
1913, 18syl 17 . . . . . . . . 9  |-  ( ph  ->  T  C_  ( Base `  R ) )
2016, 19eqsstr3d 3505 . . . . . . . 8  |-  ( ph  ->  ( Base `  H
)  C_  ( Base `  R ) )
21 mapss 7522 . . . . . . . 8  |-  ( ( ( Base `  R
)  e.  _V  /\  ( Base `  H )  C_  ( Base `  R
) )  ->  (
( Base `  H )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } )  C_  (
( Base `  R )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } ) )
2212, 20, 21sylancr 667 . . . . . . 7  |-  ( ph  ->  ( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
2322adantr 466 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
24 eqid 2429 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
25 eqid 2429 . . . . . . 7  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
26 reldmpsr 18520 . . . . . . . . . 10  |-  Rel  dom mPwSer
2726, 1, 2elbasov 15134 . . . . . . . . 9  |-  ( X  e.  B  ->  (
I  e.  _V  /\  H  e.  _V )
)
2827ad2antrl 732 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( I  e.  _V  /\  H  e.  _V )
)
2928simpld 460 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  I  e.  _V )
301, 24, 25, 2, 29psrbas 18537 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( ( Base `  H )  ^m  { f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
318, 17, 25, 9, 29psrbas 18537 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( Base `  S )  =  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
3223, 30, 313sstr4d 3513 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
3332, 5sseldd 3471 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  ( Base `  S ) )
3432, 6sseldd 3471 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
358, 9, 10, 11, 33, 34psradd 18541 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  oF ( +g  `  R
) Y ) )
3614, 10ressplusg 15198 . . . . . . 7  |-  ( T  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  H ) )
3713, 36syl 17 . . . . . 6  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  H ) )
3837adantr 466 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  R )  =  ( +g  `  H
) )
39 ofeq 6547 . . . . 5  |-  ( ( +g  `  R )  =  ( +g  `  H
)  ->  oF
( +g  `  R )  =  oF ( +g  `  H ) )
4038, 39syl 17 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  oF ( +g  `  R )  =  oF ( +g  `  H
) )
4140oveqd 6322 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  oF ( +g  `  R
) Y )  =  ( X  oF ( +g  `  H
) Y ) )
4235, 41eqtrd 2470 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  oF ( +g  `  H
) Y ) )
43 fvex 5891 . . . . 5  |-  ( Base `  U )  e.  _V
442, 43eqeltri 2513 . . . 4  |-  B  e. 
_V
45 resspsr.p . . . . 5  |-  P  =  ( Ss  B )
4645, 11ressplusg 15198 . . . 4  |-  ( B  e.  _V  ->  ( +g  `  S )  =  ( +g  `  P
) )
4744, 46mp1i 13 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  S )  =  ( +g  `  P
) )
4847oveqd 6322 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X ( +g  `  P ) Y ) )
497, 42, 483eqtr2d 2476 1  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    C_ wss 3442   `'ccnv 4853   "cima 4857   ` cfv 5601  (class class class)co 6305    oFcof 6543    ^m cmap 7480   Fincfn 7577   NNcn 10609   NN0cn0 10869   Basecbs 15084   ↾s cress 15085   +g cplusg 15152  SubRingcsubrg 17939   mPwSer cmps 18510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-tset 15171  df-subg 16765  df-ring 17717  df-subrg 17941  df-psr 18515
This theorem is referenced by:  subrgpsr  18578  ressmpladd  18616
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